Mathlib.Geometry.Pythagorean

import Mathlib.Logic.EqReasoning
import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.OrderedField.Basic
import Mathlib.Algebra.OrderedField.Square
import Mathlib.Algebra.OrderedField.ScalarIdentities
import Mathlib.LinearAlgebra.VectorSpace
import Mathlib.LinearAlgebra.InnerProduct
import Mathlib.LinearAlgebra.InnerProduct.Derived
import Mathlib.Geometry.Affine
import Mathlib.Geometry.Affine.Derived
import Mathlib.Geometry.RightTriangle.Abstract
import Mathlib.Geometry.RightTriangle.Derived
import Mathlib.Geometry.Metric.Abstract

theorem pythagorean_dist_sq_symm_from_affine_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => affine_args (@Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A)) (fun (disp_self_arg : forall (A : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A A) vzero) => fun (disp_reverse_arg : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) => fun (disp_comp_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C))) => fun (point_ext_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (h : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A B) vzero), @Eq.{p} PointCarrier A B) => fun (dist_sq_symm_arg : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A)) => fun (dist_sq_zero_iff_eq_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R), R) => dist_sq_symm_arg A B)

theorem pythagorean_dist_sq_reverse_norm_neg_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A) (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => @eq_trans.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op B A)) (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B A) (@eq_congr_arg.{v,u} Vector Scalar (fun (x : Vector) => @normSq.{u,v} Scalar Vector inner x) (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp_reverse_from_affine_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B))

theorem pythagorean_left_leg_norm_neg_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => @eq_trans.{u} Scalar (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@eq_symm.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A) (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@pythagorean_dist_sq_reverse_norm_neg_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B)) (@pythagorean_dist_sq_symm_from_affine_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B A)

theorem dist_sq_law_of_cosines_rhs_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun inner_args => fun affine_args => fun A => fun B => fun C => @eq_calc3.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))) (sub (add (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C))) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (@eq_trans.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op B C)) (@normSq.{u,v} Scalar Vector inner (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B C) (@eq_congr_arg.{v,u} Vector Scalar (fun (x : Vector) => @normSq.{u,v} Scalar Vector inner x) (@disp.{p,v} PointCarrier Vector disp_op B C) (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) (@hypotenuse_vector_eq_neg_left_add_right_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B C))) (@norm_sq_add_neg_left_from_inner_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args inner_args (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)) (@eq_congr2.{u,u,u} Scalar Scalar Scalar sub (add (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C))) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))) (@eq_congr2.{u,u,u} Scalar Scalar Scalar add (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@eq_symm.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B)) (@eq_symm.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C)) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A C))) (@Eq.refl.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))))

theorem law_of_cosines_sq_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => @dist_sq_law_of_cosines_rhs_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args inner_args affine_args A B C

theorem law_of_cosines_dist_sq_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) (sub (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C))) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ordered_args => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => @eq_calc3.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (sub (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C))) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (@square_dist_eq_dist_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args B C) (@law_of_cosines_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args vector_args inner_args affine_args A B C) (@eq_congr2.{u,u,u} Scalar Scalar Scalar sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C))) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))) (@eq_congr2.{u,u,u} Scalar Scalar Scalar add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)) (@dist_sq_eq_square_dist_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A B) (@dist_sq_eq_square_dist_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A C)) (@Eq.refl.{u} Scalar (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))))

theorem pythagorean_distance_sq_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @right_triangle_affine_additive_perp_bridge_from_rt.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args inner_args affine_args A B C h (@Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C))) (fun (hypotenuse_orientation : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))) => fun (perp_premise : @PerpVec.{u,v} Scalar zero Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) => @eq_calc3.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))) (add (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C))) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (@eq_trans.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op B C)) (@normSq.{u,v} Scalar Vector inner (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B C) (@eq_congr_arg.{v,u} Vector Scalar (fun (x : Vector) => @normSq.{u,v} Scalar Vector inner x) (@disp.{p,v} PointCarrier Vector disp_op B C) (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) hypotenuse_orientation)) (@norm_sq_add_of_perp_from_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args inner_args (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C) perp_premise) (@eq_congr2.{u,u,u} Scalar Scalar Scalar add (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@pythagorean_left_leg_norm_neg_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B) (@Eq.refl.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C))))

theorem pythagorean_theorem_sq.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @pythagorean_distance_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args vector_args inner_args affine_args A B C h

theorem pythagorean_theorem_dist_sq.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ordered_args : @OrderedFieldLawArgs.{u} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ordered_args => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @eq_calc3.{u} Scalar (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op B C)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (add (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C))) (@square_dist_eq_dist_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args B C) (@pythagorean_distance_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args vector_args inner_args affine_args A B C h) (@eq_congr2.{u,u,u} Scalar Scalar Scalar add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A B)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@sq.{u} Scalar mul (@dist.{p,u,v} Scalar sqrt_fn Vector inner PointCarrier disp_op A C)) (@dist_sq_eq_square_dist_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A B) (@dist_sq_eq_square_dist_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ordered_args inner_args A C))

theorem pythagorean_converse_sq.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (converse_target : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C))), @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C))), @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun converse_target => fun A => fun B => fun C => fun h => converse_target A B C h

theorem law_of_cosines_right_angle_specialization_from_law_packages.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @eq_trans.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (@law_of_cosines_sq_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args vector_args inner_args affine_args A B C) (@eq_trans.{u} Scalar (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (add (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (mul (@two.{u} Scalar one add) (neg (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))))) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (@eq_symm.{u} Scalar (add (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (mul (@two.{u} Scalar one add) (neg (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))))) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (sub (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (mul (@two.{u} Scalar one add) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (@law_of_cosines_scalar_rhs_from_ring_args.{u} Scalar zero one add neg sub mul ring_args (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)))) (@normalize_add_with_zero_cross_term_from_ring_args.{u} Scalar zero one add neg sub mul ring_args (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (neg (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))) (@eq_trans.{u} Scalar (neg (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C))) (neg zero) zero (@eq_congr_arg.{u,u} Scalar Scalar (fun (x : Scalar) => neg x) (@dot.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op A C)) zero (@right_triangle_legs_dot_zero_from_rt.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op A B C h)) (@neg_zero_from_ring_args.{u} Scalar zero one add neg sub mul ring_args))))

theorem law_of_cosines_right_angle_specialization.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @law_of_cosines_right_angle_specialization_from_law_packages.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args vector_args inner_args affine_args A B C h

theorem pythagorean_theorem_api_alias.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @pythagorean_theorem_sq.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args vector_args inner_args affine_args A B C h

theorem pythagorean_theorem_dependencies.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (lt_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (sqrt_fn : forall (a : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (laws : @MetricSpaceLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op), @MetricSpaceLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel lt_rel sqrt_fn Vector vzero vadd vneg smul inner PointCarrier disp_op :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun lt_rel => fun sqrt_fn => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun laws => laws